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<FONT color="green">001</FONT>    /*<a name="line.1"></a>
<FONT color="green">002</FONT>     * Licensed to the Apache Software Foundation (ASF) under one or more<a name="line.2"></a>
<FONT color="green">003</FONT>     * contributor license agreements.  See the NOTICE file distributed with<a name="line.3"></a>
<FONT color="green">004</FONT>     * this work for additional information regarding copyright ownership.<a name="line.4"></a>
<FONT color="green">005</FONT>     * The ASF licenses this file to You under the Apache License, Version 2.0<a name="line.5"></a>
<FONT color="green">006</FONT>     * (the "License"); you may not use this file except in compliance with<a name="line.6"></a>
<FONT color="green">007</FONT>     * the License.  You may obtain a copy of the License at<a name="line.7"></a>
<FONT color="green">008</FONT>     *<a name="line.8"></a>
<FONT color="green">009</FONT>     *      http://www.apache.org/licenses/LICENSE-2.0<a name="line.9"></a>
<FONT color="green">010</FONT>     *<a name="line.10"></a>
<FONT color="green">011</FONT>     * Unless required by applicable law or agreed to in writing, software<a name="line.11"></a>
<FONT color="green">012</FONT>     * distributed under the License is distributed on an "AS IS" BASIS,<a name="line.12"></a>
<FONT color="green">013</FONT>     * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.<a name="line.13"></a>
<FONT color="green">014</FONT>     * See the License for the specific language governing permissions and<a name="line.14"></a>
<FONT color="green">015</FONT>     * limitations under the License.<a name="line.15"></a>
<FONT color="green">016</FONT>     */<a name="line.16"></a>
<FONT color="green">017</FONT>    package org.apache.commons.math3.analysis.solvers;<a name="line.17"></a>
<FONT color="green">018</FONT>    <a name="line.18"></a>
<FONT color="green">019</FONT>    import org.apache.commons.math3.complex.Complex;<a name="line.19"></a>
<FONT color="green">020</FONT>    import org.apache.commons.math3.complex.ComplexUtils;<a name="line.20"></a>
<FONT color="green">021</FONT>    import org.apache.commons.math3.analysis.polynomials.PolynomialFunction;<a name="line.21"></a>
<FONT color="green">022</FONT>    import org.apache.commons.math3.exception.NoBracketingException;<a name="line.22"></a>
<FONT color="green">023</FONT>    import org.apache.commons.math3.exception.NullArgumentException;<a name="line.23"></a>
<FONT color="green">024</FONT>    import org.apache.commons.math3.exception.NoDataException;<a name="line.24"></a>
<FONT color="green">025</FONT>    import org.apache.commons.math3.exception.TooManyEvaluationsException;<a name="line.25"></a>
<FONT color="green">026</FONT>    import org.apache.commons.math3.exception.NumberIsTooLargeException;<a name="line.26"></a>
<FONT color="green">027</FONT>    import org.apache.commons.math3.exception.util.LocalizedFormats;<a name="line.27"></a>
<FONT color="green">028</FONT>    import org.apache.commons.math3.util.FastMath;<a name="line.28"></a>
<FONT color="green">029</FONT>    <a name="line.29"></a>
<FONT color="green">030</FONT>    /**<a name="line.30"></a>
<FONT color="green">031</FONT>     * Implements the &lt;a href="http://mathworld.wolfram.com/LaguerresMethod.html"&gt;<a name="line.31"></a>
<FONT color="green">032</FONT>     * Laguerre's Method&lt;/a&gt; for root finding of real coefficient polynomials.<a name="line.32"></a>
<FONT color="green">033</FONT>     * For reference, see<a name="line.33"></a>
<FONT color="green">034</FONT>     * &lt;quote&gt;<a name="line.34"></a>
<FONT color="green">035</FONT>     *  &lt;b&gt;A First Course in Numerical Analysis&lt;/b&gt;<a name="line.35"></a>
<FONT color="green">036</FONT>     *  ISBN 048641454X, chapter 8.<a name="line.36"></a>
<FONT color="green">037</FONT>     * &lt;/quote&gt;<a name="line.37"></a>
<FONT color="green">038</FONT>     * Laguerre's method is global in the sense that it can start with any initial<a name="line.38"></a>
<FONT color="green">039</FONT>     * approximation and be able to solve all roots from that point.<a name="line.39"></a>
<FONT color="green">040</FONT>     * The algorithm requires a bracketing condition.<a name="line.40"></a>
<FONT color="green">041</FONT>     *<a name="line.41"></a>
<FONT color="green">042</FONT>     * @version $Id: LaguerreSolver.java 1422195 2012-12-15 06:45:18Z psteitz $<a name="line.42"></a>
<FONT color="green">043</FONT>     * @since 1.2<a name="line.43"></a>
<FONT color="green">044</FONT>     */<a name="line.44"></a>
<FONT color="green">045</FONT>    public class LaguerreSolver extends AbstractPolynomialSolver {<a name="line.45"></a>
<FONT color="green">046</FONT>        /** Default absolute accuracy. */<a name="line.46"></a>
<FONT color="green">047</FONT>        private static final double DEFAULT_ABSOLUTE_ACCURACY = 1e-6;<a name="line.47"></a>
<FONT color="green">048</FONT>        /** Complex solver. */<a name="line.48"></a>
<FONT color="green">049</FONT>        private final ComplexSolver complexSolver = new ComplexSolver();<a name="line.49"></a>
<FONT color="green">050</FONT>    <a name="line.50"></a>
<FONT color="green">051</FONT>        /**<a name="line.51"></a>
<FONT color="green">052</FONT>         * Construct a solver with default accuracy (1e-6).<a name="line.52"></a>
<FONT color="green">053</FONT>         */<a name="line.53"></a>
<FONT color="green">054</FONT>        public LaguerreSolver() {<a name="line.54"></a>
<FONT color="green">055</FONT>            this(DEFAULT_ABSOLUTE_ACCURACY);<a name="line.55"></a>
<FONT color="green">056</FONT>        }<a name="line.56"></a>
<FONT color="green">057</FONT>        /**<a name="line.57"></a>
<FONT color="green">058</FONT>         * Construct a solver.<a name="line.58"></a>
<FONT color="green">059</FONT>         *<a name="line.59"></a>
<FONT color="green">060</FONT>         * @param absoluteAccuracy Absolute accuracy.<a name="line.60"></a>
<FONT color="green">061</FONT>         */<a name="line.61"></a>
<FONT color="green">062</FONT>        public LaguerreSolver(double absoluteAccuracy) {<a name="line.62"></a>
<FONT color="green">063</FONT>            super(absoluteAccuracy);<a name="line.63"></a>
<FONT color="green">064</FONT>        }<a name="line.64"></a>
<FONT color="green">065</FONT>        /**<a name="line.65"></a>
<FONT color="green">066</FONT>         * Construct a solver.<a name="line.66"></a>
<FONT color="green">067</FONT>         *<a name="line.67"></a>
<FONT color="green">068</FONT>         * @param relativeAccuracy Relative accuracy.<a name="line.68"></a>
<FONT color="green">069</FONT>         * @param absoluteAccuracy Absolute accuracy.<a name="line.69"></a>
<FONT color="green">070</FONT>         */<a name="line.70"></a>
<FONT color="green">071</FONT>        public LaguerreSolver(double relativeAccuracy,<a name="line.71"></a>
<FONT color="green">072</FONT>                              double absoluteAccuracy) {<a name="line.72"></a>
<FONT color="green">073</FONT>            super(relativeAccuracy, absoluteAccuracy);<a name="line.73"></a>
<FONT color="green">074</FONT>        }<a name="line.74"></a>
<FONT color="green">075</FONT>        /**<a name="line.75"></a>
<FONT color="green">076</FONT>         * Construct a solver.<a name="line.76"></a>
<FONT color="green">077</FONT>         *<a name="line.77"></a>
<FONT color="green">078</FONT>         * @param relativeAccuracy Relative accuracy.<a name="line.78"></a>
<FONT color="green">079</FONT>         * @param absoluteAccuracy Absolute accuracy.<a name="line.79"></a>
<FONT color="green">080</FONT>         * @param functionValueAccuracy Function value accuracy.<a name="line.80"></a>
<FONT color="green">081</FONT>         */<a name="line.81"></a>
<FONT color="green">082</FONT>        public LaguerreSolver(double relativeAccuracy,<a name="line.82"></a>
<FONT color="green">083</FONT>                              double absoluteAccuracy,<a name="line.83"></a>
<FONT color="green">084</FONT>                              double functionValueAccuracy) {<a name="line.84"></a>
<FONT color="green">085</FONT>            super(relativeAccuracy, absoluteAccuracy, functionValueAccuracy);<a name="line.85"></a>
<FONT color="green">086</FONT>        }<a name="line.86"></a>
<FONT color="green">087</FONT>    <a name="line.87"></a>
<FONT color="green">088</FONT>        /**<a name="line.88"></a>
<FONT color="green">089</FONT>         * {@inheritDoc}<a name="line.89"></a>
<FONT color="green">090</FONT>         */<a name="line.90"></a>
<FONT color="green">091</FONT>        @Override<a name="line.91"></a>
<FONT color="green">092</FONT>        public double doSolve()<a name="line.92"></a>
<FONT color="green">093</FONT>            throws TooManyEvaluationsException,<a name="line.93"></a>
<FONT color="green">094</FONT>                   NumberIsTooLargeException,<a name="line.94"></a>
<FONT color="green">095</FONT>                   NoBracketingException {<a name="line.95"></a>
<FONT color="green">096</FONT>            final double min = getMin();<a name="line.96"></a>
<FONT color="green">097</FONT>            final double max = getMax();<a name="line.97"></a>
<FONT color="green">098</FONT>            final double initial = getStartValue();<a name="line.98"></a>
<FONT color="green">099</FONT>            final double functionValueAccuracy = getFunctionValueAccuracy();<a name="line.99"></a>
<FONT color="green">100</FONT>    <a name="line.100"></a>
<FONT color="green">101</FONT>            verifySequence(min, initial, max);<a name="line.101"></a>
<FONT color="green">102</FONT>    <a name="line.102"></a>
<FONT color="green">103</FONT>            // Return the initial guess if it is good enough.<a name="line.103"></a>
<FONT color="green">104</FONT>            final double yInitial = computeObjectiveValue(initial);<a name="line.104"></a>
<FONT color="green">105</FONT>            if (FastMath.abs(yInitial) &lt;= functionValueAccuracy) {<a name="line.105"></a>
<FONT color="green">106</FONT>                return initial;<a name="line.106"></a>
<FONT color="green">107</FONT>            }<a name="line.107"></a>
<FONT color="green">108</FONT>    <a name="line.108"></a>
<FONT color="green">109</FONT>            // Return the first endpoint if it is good enough.<a name="line.109"></a>
<FONT color="green">110</FONT>            final double yMin = computeObjectiveValue(min);<a name="line.110"></a>
<FONT color="green">111</FONT>            if (FastMath.abs(yMin) &lt;= functionValueAccuracy) {<a name="line.111"></a>
<FONT color="green">112</FONT>                return min;<a name="line.112"></a>
<FONT color="green">113</FONT>            }<a name="line.113"></a>
<FONT color="green">114</FONT>    <a name="line.114"></a>
<FONT color="green">115</FONT>            // Reduce interval if min and initial bracket the root.<a name="line.115"></a>
<FONT color="green">116</FONT>            if (yInitial * yMin &lt; 0) {<a name="line.116"></a>
<FONT color="green">117</FONT>                return laguerre(min, initial, yMin, yInitial);<a name="line.117"></a>
<FONT color="green">118</FONT>            }<a name="line.118"></a>
<FONT color="green">119</FONT>    <a name="line.119"></a>
<FONT color="green">120</FONT>            // Return the second endpoint if it is good enough.<a name="line.120"></a>
<FONT color="green">121</FONT>            final double yMax = computeObjectiveValue(max);<a name="line.121"></a>
<FONT color="green">122</FONT>            if (FastMath.abs(yMax) &lt;= functionValueAccuracy) {<a name="line.122"></a>
<FONT color="green">123</FONT>                return max;<a name="line.123"></a>
<FONT color="green">124</FONT>            }<a name="line.124"></a>
<FONT color="green">125</FONT>    <a name="line.125"></a>
<FONT color="green">126</FONT>            // Reduce interval if initial and max bracket the root.<a name="line.126"></a>
<FONT color="green">127</FONT>            if (yInitial * yMax &lt; 0) {<a name="line.127"></a>
<FONT color="green">128</FONT>                return laguerre(initial, max, yInitial, yMax);<a name="line.128"></a>
<FONT color="green">129</FONT>            }<a name="line.129"></a>
<FONT color="green">130</FONT>    <a name="line.130"></a>
<FONT color="green">131</FONT>            throw new NoBracketingException(min, max, yMin, yMax);<a name="line.131"></a>
<FONT color="green">132</FONT>        }<a name="line.132"></a>
<FONT color="green">133</FONT>    <a name="line.133"></a>
<FONT color="green">134</FONT>        /**<a name="line.134"></a>
<FONT color="green">135</FONT>         * Find a real root in the given interval.<a name="line.135"></a>
<FONT color="green">136</FONT>         *<a name="line.136"></a>
<FONT color="green">137</FONT>         * Despite the bracketing condition, the root returned by<a name="line.137"></a>
<FONT color="green">138</FONT>         * {@link LaguerreSolver.ComplexSolver#solve(Complex[],Complex)} may<a name="line.138"></a>
<FONT color="green">139</FONT>         * not be a real zero inside {@code [min, max]}.<a name="line.139"></a>
<FONT color="green">140</FONT>         * For example, &lt;code&gt;p(x) = x&lt;sup&gt;3&lt;/sup&gt; + 1,&lt;/code&gt;<a name="line.140"></a>
<FONT color="green">141</FONT>         * with {@code min = -2}, {@code max = 2}, {@code initial = 0}.<a name="line.141"></a>
<FONT color="green">142</FONT>         * When it occurs, this code calls<a name="line.142"></a>
<FONT color="green">143</FONT>         * {@link LaguerreSolver.ComplexSolver#solveAll(Complex[],Complex)}<a name="line.143"></a>
<FONT color="green">144</FONT>         * in order to obtain all roots and picks up one real root.<a name="line.144"></a>
<FONT color="green">145</FONT>         *<a name="line.145"></a>
<FONT color="green">146</FONT>         * @param lo Lower bound of the search interval.<a name="line.146"></a>
<FONT color="green">147</FONT>         * @param hi Higher bound of the search interval.<a name="line.147"></a>
<FONT color="green">148</FONT>         * @param fLo Function value at the lower bound of the search interval.<a name="line.148"></a>
<FONT color="green">149</FONT>         * @param fHi Function value at the higher bound of the search interval.<a name="line.149"></a>
<FONT color="green">150</FONT>         * @return the point at which the function value is zero.<a name="line.150"></a>
<FONT color="green">151</FONT>         * @deprecated This method should not be part of the public API: It will<a name="line.151"></a>
<FONT color="green">152</FONT>         * be made private in version 4.0.<a name="line.152"></a>
<FONT color="green">153</FONT>         */<a name="line.153"></a>
<FONT color="green">154</FONT>        @Deprecated<a name="line.154"></a>
<FONT color="green">155</FONT>        public double laguerre(double lo, double hi,<a name="line.155"></a>
<FONT color="green">156</FONT>                               double fLo, double fHi) {<a name="line.156"></a>
<FONT color="green">157</FONT>            final Complex c[] = ComplexUtils.convertToComplex(getCoefficients());<a name="line.157"></a>
<FONT color="green">158</FONT>    <a name="line.158"></a>
<FONT color="green">159</FONT>            final Complex initial = new Complex(0.5 * (lo + hi), 0);<a name="line.159"></a>
<FONT color="green">160</FONT>            final Complex z = complexSolver.solve(c, initial);<a name="line.160"></a>
<FONT color="green">161</FONT>            if (complexSolver.isRoot(lo, hi, z)) {<a name="line.161"></a>
<FONT color="green">162</FONT>                return z.getReal();<a name="line.162"></a>
<FONT color="green">163</FONT>            } else {<a name="line.163"></a>
<FONT color="green">164</FONT>                double r = Double.NaN;<a name="line.164"></a>
<FONT color="green">165</FONT>                // Solve all roots and select the one we are seeking.<a name="line.165"></a>
<FONT color="green">166</FONT>                Complex[] root = complexSolver.solveAll(c, initial);<a name="line.166"></a>
<FONT color="green">167</FONT>                for (int i = 0; i &lt; root.length; i++) {<a name="line.167"></a>
<FONT color="green">168</FONT>                    if (complexSolver.isRoot(lo, hi, root[i])) {<a name="line.168"></a>
<FONT color="green">169</FONT>                        r = root[i].getReal();<a name="line.169"></a>
<FONT color="green">170</FONT>                        break;<a name="line.170"></a>
<FONT color="green">171</FONT>                    }<a name="line.171"></a>
<FONT color="green">172</FONT>                }<a name="line.172"></a>
<FONT color="green">173</FONT>                return r;<a name="line.173"></a>
<FONT color="green">174</FONT>            }<a name="line.174"></a>
<FONT color="green">175</FONT>        }<a name="line.175"></a>
<FONT color="green">176</FONT>    <a name="line.176"></a>
<FONT color="green">177</FONT>        /**<a name="line.177"></a>
<FONT color="green">178</FONT>         * Find all complex roots for the polynomial with the given<a name="line.178"></a>
<FONT color="green">179</FONT>         * coefficients, starting from the given initial value.<a name="line.179"></a>
<FONT color="green">180</FONT>         * &lt;br/&gt;<a name="line.180"></a>
<FONT color="green">181</FONT>         * Note: This method is not part of the API of {@link BaseUnivariateSolver}.<a name="line.181"></a>
<FONT color="green">182</FONT>         *<a name="line.182"></a>
<FONT color="green">183</FONT>         * @param coefficients Polynomial coefficients.<a name="line.183"></a>
<FONT color="green">184</FONT>         * @param initial Start value.<a name="line.184"></a>
<FONT color="green">185</FONT>         * @return the point at which the function value is zero.<a name="line.185"></a>
<FONT color="green">186</FONT>         * @throws org.apache.commons.math3.exception.TooManyEvaluationsException<a name="line.186"></a>
<FONT color="green">187</FONT>         * if the maximum number of evaluations is exceeded.<a name="line.187"></a>
<FONT color="green">188</FONT>         * @throws NullArgumentException if the {@code coefficients} is<a name="line.188"></a>
<FONT color="green">189</FONT>         * {@code null}.<a name="line.189"></a>
<FONT color="green">190</FONT>         * @throws NoDataException if the {@code coefficients} array is empty.<a name="line.190"></a>
<FONT color="green">191</FONT>         * @since 3.1<a name="line.191"></a>
<FONT color="green">192</FONT>         */<a name="line.192"></a>
<FONT color="green">193</FONT>        public Complex[] solveAllComplex(double[] coefficients,<a name="line.193"></a>
<FONT color="green">194</FONT>                                         double initial)<a name="line.194"></a>
<FONT color="green">195</FONT>            throws NullArgumentException,<a name="line.195"></a>
<FONT color="green">196</FONT>                   NoDataException,<a name="line.196"></a>
<FONT color="green">197</FONT>                   TooManyEvaluationsException {<a name="line.197"></a>
<FONT color="green">198</FONT>            setup(Integer.MAX_VALUE,<a name="line.198"></a>
<FONT color="green">199</FONT>                  new PolynomialFunction(coefficients),<a name="line.199"></a>
<FONT color="green">200</FONT>                  Double.NEGATIVE_INFINITY,<a name="line.200"></a>
<FONT color="green">201</FONT>                  Double.POSITIVE_INFINITY,<a name="line.201"></a>
<FONT color="green">202</FONT>                  initial);<a name="line.202"></a>
<FONT color="green">203</FONT>            return complexSolver.solveAll(ComplexUtils.convertToComplex(coefficients),<a name="line.203"></a>
<FONT color="green">204</FONT>                                          new Complex(initial, 0d));<a name="line.204"></a>
<FONT color="green">205</FONT>        }<a name="line.205"></a>
<FONT color="green">206</FONT>    <a name="line.206"></a>
<FONT color="green">207</FONT>        /**<a name="line.207"></a>
<FONT color="green">208</FONT>         * Find a complex root for the polynomial with the given coefficients,<a name="line.208"></a>
<FONT color="green">209</FONT>         * starting from the given initial value.<a name="line.209"></a>
<FONT color="green">210</FONT>         * &lt;br/&gt;<a name="line.210"></a>
<FONT color="green">211</FONT>         * Note: This method is not part of the API of {@link BaseUnivariateSolver}.<a name="line.211"></a>
<FONT color="green">212</FONT>         *<a name="line.212"></a>
<FONT color="green">213</FONT>         * @param coefficients Polynomial coefficients.<a name="line.213"></a>
<FONT color="green">214</FONT>         * @param initial Start value.<a name="line.214"></a>
<FONT color="green">215</FONT>         * @return the point at which the function value is zero.<a name="line.215"></a>
<FONT color="green">216</FONT>         * @throws org.apache.commons.math3.exception.TooManyEvaluationsException<a name="line.216"></a>
<FONT color="green">217</FONT>         * if the maximum number of evaluations is exceeded.<a name="line.217"></a>
<FONT color="green">218</FONT>         * @throws NullArgumentException if the {@code coefficients} is<a name="line.218"></a>
<FONT color="green">219</FONT>         * {@code null}.<a name="line.219"></a>
<FONT color="green">220</FONT>         * @throws NoDataException if the {@code coefficients} array is empty.<a name="line.220"></a>
<FONT color="green">221</FONT>         * @since 3.1<a name="line.221"></a>
<FONT color="green">222</FONT>         */<a name="line.222"></a>
<FONT color="green">223</FONT>        public Complex solveComplex(double[] coefficients,<a name="line.223"></a>
<FONT color="green">224</FONT>                                    double initial)<a name="line.224"></a>
<FONT color="green">225</FONT>            throws NullArgumentException,<a name="line.225"></a>
<FONT color="green">226</FONT>                   NoDataException,<a name="line.226"></a>
<FONT color="green">227</FONT>                   TooManyEvaluationsException {<a name="line.227"></a>
<FONT color="green">228</FONT>            setup(Integer.MAX_VALUE,<a name="line.228"></a>
<FONT color="green">229</FONT>                  new PolynomialFunction(coefficients),<a name="line.229"></a>
<FONT color="green">230</FONT>                  Double.NEGATIVE_INFINITY,<a name="line.230"></a>
<FONT color="green">231</FONT>                  Double.POSITIVE_INFINITY,<a name="line.231"></a>
<FONT color="green">232</FONT>                  initial);<a name="line.232"></a>
<FONT color="green">233</FONT>            return complexSolver.solve(ComplexUtils.convertToComplex(coefficients),<a name="line.233"></a>
<FONT color="green">234</FONT>                                       new Complex(initial, 0d));<a name="line.234"></a>
<FONT color="green">235</FONT>        }<a name="line.235"></a>
<FONT color="green">236</FONT>    <a name="line.236"></a>
<FONT color="green">237</FONT>        /**<a name="line.237"></a>
<FONT color="green">238</FONT>         * Class for searching all (complex) roots.<a name="line.238"></a>
<FONT color="green">239</FONT>         */<a name="line.239"></a>
<FONT color="green">240</FONT>        private class ComplexSolver {<a name="line.240"></a>
<FONT color="green">241</FONT>            /**<a name="line.241"></a>
<FONT color="green">242</FONT>             * Check whether the given complex root is actually a real zero<a name="line.242"></a>
<FONT color="green">243</FONT>             * in the given interval, within the solver tolerance level.<a name="line.243"></a>
<FONT color="green">244</FONT>             *<a name="line.244"></a>
<FONT color="green">245</FONT>             * @param min Lower bound for the interval.<a name="line.245"></a>
<FONT color="green">246</FONT>             * @param max Upper bound for the interval.<a name="line.246"></a>
<FONT color="green">247</FONT>             * @param z Complex root.<a name="line.247"></a>
<FONT color="green">248</FONT>             * @return {@code true} if z is a real zero.<a name="line.248"></a>
<FONT color="green">249</FONT>             */<a name="line.249"></a>
<FONT color="green">250</FONT>            public boolean isRoot(double min, double max, Complex z) {<a name="line.250"></a>
<FONT color="green">251</FONT>                if (isSequence(min, z.getReal(), max)) {<a name="line.251"></a>
<FONT color="green">252</FONT>                    double tolerance = FastMath.max(getRelativeAccuracy() * z.abs(), getAbsoluteAccuracy());<a name="line.252"></a>
<FONT color="green">253</FONT>                    return (FastMath.abs(z.getImaginary()) &lt;= tolerance) ||<a name="line.253"></a>
<FONT color="green">254</FONT>                         (z.abs() &lt;= getFunctionValueAccuracy());<a name="line.254"></a>
<FONT color="green">255</FONT>                }<a name="line.255"></a>
<FONT color="green">256</FONT>                return false;<a name="line.256"></a>
<FONT color="green">257</FONT>            }<a name="line.257"></a>
<FONT color="green">258</FONT>    <a name="line.258"></a>
<FONT color="green">259</FONT>            /**<a name="line.259"></a>
<FONT color="green">260</FONT>             * Find all complex roots for the polynomial with the given<a name="line.260"></a>
<FONT color="green">261</FONT>             * coefficients, starting from the given initial value.<a name="line.261"></a>
<FONT color="green">262</FONT>             *<a name="line.262"></a>
<FONT color="green">263</FONT>             * @param coefficients Polynomial coefficients.<a name="line.263"></a>
<FONT color="green">264</FONT>             * @param initial Start value.<a name="line.264"></a>
<FONT color="green">265</FONT>             * @return the point at which the function value is zero.<a name="line.265"></a>
<FONT color="green">266</FONT>             * @throws org.apache.commons.math3.exception.TooManyEvaluationsException<a name="line.266"></a>
<FONT color="green">267</FONT>             * if the maximum number of evaluations is exceeded.<a name="line.267"></a>
<FONT color="green">268</FONT>             * @throws NullArgumentException if the {@code coefficients} is<a name="line.268"></a>
<FONT color="green">269</FONT>             * {@code null}.<a name="line.269"></a>
<FONT color="green">270</FONT>             * @throws NoDataException if the {@code coefficients} array is empty.<a name="line.270"></a>
<FONT color="green">271</FONT>             */<a name="line.271"></a>
<FONT color="green">272</FONT>            public Complex[] solveAll(Complex coefficients[], Complex initial)<a name="line.272"></a>
<FONT color="green">273</FONT>                throws NullArgumentException,<a name="line.273"></a>
<FONT color="green">274</FONT>                       NoDataException,<a name="line.274"></a>
<FONT color="green">275</FONT>                       TooManyEvaluationsException {<a name="line.275"></a>
<FONT color="green">276</FONT>                if (coefficients == null) {<a name="line.276"></a>
<FONT color="green">277</FONT>                    throw new NullArgumentException();<a name="line.277"></a>
<FONT color="green">278</FONT>                }<a name="line.278"></a>
<FONT color="green">279</FONT>                final int n = coefficients.length - 1;<a name="line.279"></a>
<FONT color="green">280</FONT>                if (n == 0) {<a name="line.280"></a>
<FONT color="green">281</FONT>                    throw new NoDataException(LocalizedFormats.POLYNOMIAL);<a name="line.281"></a>
<FONT color="green">282</FONT>                }<a name="line.282"></a>
<FONT color="green">283</FONT>                // Coefficients for deflated polynomial.<a name="line.283"></a>
<FONT color="green">284</FONT>                final Complex c[] = new Complex[n + 1];<a name="line.284"></a>
<FONT color="green">285</FONT>                for (int i = 0; i &lt;= n; i++) {<a name="line.285"></a>
<FONT color="green">286</FONT>                    c[i] = coefficients[i];<a name="line.286"></a>
<FONT color="green">287</FONT>                }<a name="line.287"></a>
<FONT color="green">288</FONT>    <a name="line.288"></a>
<FONT color="green">289</FONT>                // Solve individual roots successively.<a name="line.289"></a>
<FONT color="green">290</FONT>                final Complex root[] = new Complex[n];<a name="line.290"></a>
<FONT color="green">291</FONT>                for (int i = 0; i &lt; n; i++) {<a name="line.291"></a>
<FONT color="green">292</FONT>                    final Complex subarray[] = new Complex[n - i + 1];<a name="line.292"></a>
<FONT color="green">293</FONT>                    System.arraycopy(c, 0, subarray, 0, subarray.length);<a name="line.293"></a>
<FONT color="green">294</FONT>                    root[i] = solve(subarray, initial);<a name="line.294"></a>
<FONT color="green">295</FONT>                    // Polynomial deflation using synthetic division.<a name="line.295"></a>
<FONT color="green">296</FONT>                    Complex newc = c[n - i];<a name="line.296"></a>
<FONT color="green">297</FONT>                    Complex oldc = null;<a name="line.297"></a>
<FONT color="green">298</FONT>                    for (int j = n - i - 1; j &gt;= 0; j--) {<a name="line.298"></a>
<FONT color="green">299</FONT>                        oldc = c[j];<a name="line.299"></a>
<FONT color="green">300</FONT>                        c[j] = newc;<a name="line.300"></a>
<FONT color="green">301</FONT>                        newc = oldc.add(newc.multiply(root[i]));<a name="line.301"></a>
<FONT color="green">302</FONT>                    }<a name="line.302"></a>
<FONT color="green">303</FONT>                }<a name="line.303"></a>
<FONT color="green">304</FONT>    <a name="line.304"></a>
<FONT color="green">305</FONT>                return root;<a name="line.305"></a>
<FONT color="green">306</FONT>            }<a name="line.306"></a>
<FONT color="green">307</FONT>    <a name="line.307"></a>
<FONT color="green">308</FONT>            /**<a name="line.308"></a>
<FONT color="green">309</FONT>             * Find a complex root for the polynomial with the given coefficients,<a name="line.309"></a>
<FONT color="green">310</FONT>             * starting from the given initial value.<a name="line.310"></a>
<FONT color="green">311</FONT>             *<a name="line.311"></a>
<FONT color="green">312</FONT>             * @param coefficients Polynomial coefficients.<a name="line.312"></a>
<FONT color="green">313</FONT>             * @param initial Start value.<a name="line.313"></a>
<FONT color="green">314</FONT>             * @return the point at which the function value is zero.<a name="line.314"></a>
<FONT color="green">315</FONT>             * @throws org.apache.commons.math3.exception.TooManyEvaluationsException<a name="line.315"></a>
<FONT color="green">316</FONT>             * if the maximum number of evaluations is exceeded.<a name="line.316"></a>
<FONT color="green">317</FONT>             * @throws NullArgumentException if the {@code coefficients} is<a name="line.317"></a>
<FONT color="green">318</FONT>             * {@code null}.<a name="line.318"></a>
<FONT color="green">319</FONT>             * @throws NoDataException if the {@code coefficients} array is empty.<a name="line.319"></a>
<FONT color="green">320</FONT>             */<a name="line.320"></a>
<FONT color="green">321</FONT>            public Complex solve(Complex coefficients[], Complex initial)<a name="line.321"></a>
<FONT color="green">322</FONT>                throws NullArgumentException,<a name="line.322"></a>
<FONT color="green">323</FONT>                       NoDataException,<a name="line.323"></a>
<FONT color="green">324</FONT>                       TooManyEvaluationsException {<a name="line.324"></a>
<FONT color="green">325</FONT>                if (coefficients == null) {<a name="line.325"></a>
<FONT color="green">326</FONT>                    throw new NullArgumentException();<a name="line.326"></a>
<FONT color="green">327</FONT>                }<a name="line.327"></a>
<FONT color="green">328</FONT>    <a name="line.328"></a>
<FONT color="green">329</FONT>                final int n = coefficients.length - 1;<a name="line.329"></a>
<FONT color="green">330</FONT>                if (n == 0) {<a name="line.330"></a>
<FONT color="green">331</FONT>                    throw new NoDataException(LocalizedFormats.POLYNOMIAL);<a name="line.331"></a>
<FONT color="green">332</FONT>                }<a name="line.332"></a>
<FONT color="green">333</FONT>    <a name="line.333"></a>
<FONT color="green">334</FONT>                final double absoluteAccuracy = getAbsoluteAccuracy();<a name="line.334"></a>
<FONT color="green">335</FONT>                final double relativeAccuracy = getRelativeAccuracy();<a name="line.335"></a>
<FONT color="green">336</FONT>                final double functionValueAccuracy = getFunctionValueAccuracy();<a name="line.336"></a>
<FONT color="green">337</FONT>    <a name="line.337"></a>
<FONT color="green">338</FONT>                final Complex nC  = new Complex(n, 0);<a name="line.338"></a>
<FONT color="green">339</FONT>                final Complex n1C = new Complex(n - 1, 0);<a name="line.339"></a>
<FONT color="green">340</FONT>    <a name="line.340"></a>
<FONT color="green">341</FONT>                Complex z = initial;<a name="line.341"></a>
<FONT color="green">342</FONT>                Complex oldz = new Complex(Double.POSITIVE_INFINITY,<a name="line.342"></a>
<FONT color="green">343</FONT>                                           Double.POSITIVE_INFINITY);<a name="line.343"></a>
<FONT color="green">344</FONT>                while (true) {<a name="line.344"></a>
<FONT color="green">345</FONT>                    // Compute pv (polynomial value), dv (derivative value), and<a name="line.345"></a>
<FONT color="green">346</FONT>                    // d2v (second derivative value) simultaneously.<a name="line.346"></a>
<FONT color="green">347</FONT>                    Complex pv = coefficients[n];<a name="line.347"></a>
<FONT color="green">348</FONT>                    Complex dv = Complex.ZERO;<a name="line.348"></a>
<FONT color="green">349</FONT>                    Complex d2v = Complex.ZERO;<a name="line.349"></a>
<FONT color="green">350</FONT>                    for (int j = n-1; j &gt;= 0; j--) {<a name="line.350"></a>
<FONT color="green">351</FONT>                        d2v = dv.add(z.multiply(d2v));<a name="line.351"></a>
<FONT color="green">352</FONT>                        dv = pv.add(z.multiply(dv));<a name="line.352"></a>
<FONT color="green">353</FONT>                        pv = coefficients[j].add(z.multiply(pv));<a name="line.353"></a>
<FONT color="green">354</FONT>                    }<a name="line.354"></a>
<FONT color="green">355</FONT>                    d2v = d2v.multiply(new Complex(2.0, 0.0));<a name="line.355"></a>
<FONT color="green">356</FONT>    <a name="line.356"></a>
<FONT color="green">357</FONT>                    // Check for convergence.<a name="line.357"></a>
<FONT color="green">358</FONT>                    final double tolerance = FastMath.max(relativeAccuracy * z.abs(),<a name="line.358"></a>
<FONT color="green">359</FONT>                                                          absoluteAccuracy);<a name="line.359"></a>
<FONT color="green">360</FONT>                    if ((z.subtract(oldz)).abs() &lt;= tolerance) {<a name="line.360"></a>
<FONT color="green">361</FONT>                        return z;<a name="line.361"></a>
<FONT color="green">362</FONT>                    }<a name="line.362"></a>
<FONT color="green">363</FONT>                    if (pv.abs() &lt;= functionValueAccuracy) {<a name="line.363"></a>
<FONT color="green">364</FONT>                        return z;<a name="line.364"></a>
<FONT color="green">365</FONT>                    }<a name="line.365"></a>
<FONT color="green">366</FONT>    <a name="line.366"></a>
<FONT color="green">367</FONT>                    // Now pv != 0, calculate the new approximation.<a name="line.367"></a>
<FONT color="green">368</FONT>                    final Complex G = dv.divide(pv);<a name="line.368"></a>
<FONT color="green">369</FONT>                    final Complex G2 = G.multiply(G);<a name="line.369"></a>
<FONT color="green">370</FONT>                    final Complex H = G2.subtract(d2v.divide(pv));<a name="line.370"></a>
<FONT color="green">371</FONT>                    final Complex delta = n1C.multiply((nC.multiply(H)).subtract(G2));<a name="line.371"></a>
<FONT color="green">372</FONT>                    // Choose a denominator larger in magnitude.<a name="line.372"></a>
<FONT color="green">373</FONT>                    final Complex deltaSqrt = delta.sqrt();<a name="line.373"></a>
<FONT color="green">374</FONT>                    final Complex dplus = G.add(deltaSqrt);<a name="line.374"></a>
<FONT color="green">375</FONT>                    final Complex dminus = G.subtract(deltaSqrt);<a name="line.375"></a>
<FONT color="green">376</FONT>                    final Complex denominator = dplus.abs() &gt; dminus.abs() ? dplus : dminus;<a name="line.376"></a>
<FONT color="green">377</FONT>                    // Perturb z if denominator is zero, for instance,<a name="line.377"></a>
<FONT color="green">378</FONT>                    // p(x) = x^3 + 1, z = 0.<a name="line.378"></a>
<FONT color="green">379</FONT>                    if (denominator.equals(new Complex(0.0, 0.0))) {<a name="line.379"></a>
<FONT color="green">380</FONT>                        z = z.add(new Complex(absoluteAccuracy, absoluteAccuracy));<a name="line.380"></a>
<FONT color="green">381</FONT>                        oldz = new Complex(Double.POSITIVE_INFINITY,<a name="line.381"></a>
<FONT color="green">382</FONT>                                           Double.POSITIVE_INFINITY);<a name="line.382"></a>
<FONT color="green">383</FONT>                    } else {<a name="line.383"></a>
<FONT color="green">384</FONT>                        oldz = z;<a name="line.384"></a>
<FONT color="green">385</FONT>                        z = z.subtract(nC.divide(denominator));<a name="line.385"></a>
<FONT color="green">386</FONT>                    }<a name="line.386"></a>
<FONT color="green">387</FONT>                    incrementEvaluationCount();<a name="line.387"></a>
<FONT color="green">388</FONT>                }<a name="line.388"></a>
<FONT color="green">389</FONT>            }<a name="line.389"></a>
<FONT color="green">390</FONT>        }<a name="line.390"></a>
<FONT color="green">391</FONT>    }<a name="line.391"></a>




























































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